Normed versus topological groups: Dichotomy and duality

Normed groups 91left-shift, not in general even continuous. The alternative T (x, y) = xy −1 xy −1 introducesshift operations to the left of the second x.7. The Kestelman-Borwein-Ditor Theorem: a bitopologicalapproachIn this section we develop a bi-topological approach to a generalization of the KBDTheorem (Th. 1.1). An alternative approach is given in the next section. Let (X, S, m)be a probability space which is totally finite. Let m ∗ denote the outer measurem ∗ (E) := inf{m(F ) : E ⊂ F ∈ S}.Let the family {K n (x) : x ∈ X} ⊂ S satisfy (i) x ∈ K n (x), (ii) m(K n (x)) → 0.Relative to a fixed family {K n (x) : x ∈ X} define the upper and lower (outer) density atx of any set E byD ∗ (E, x) = sup lim sup n m ∗ (E ∩ K n (x))/m(K n (x)),D ∗ (E, x) = inf lim inf n m ∗ (E ∩ K n (x))/m(K n (x)).By definition D ∗ (E, x) ≥ D ∗ (E, x). When equality holds, one says that the densityof E exists at x, and the common value is denoted by D ∗ (E, x). If E is measurable thestar associated with the outer measure m ∗ is omitted. If the density is 1 at x, then xis a density point; if the density is 0 at x, then x is a dispersion point of E. Say that a(weak) density theorem holds for {K n (x) : x ∈ X} if for every set (every measurable set)A almost every point of A is a density (an outer density) point of A. Martin [Mar] showsthat the familyU = {U : D ∗ (X\U, x) = 0, for all x ∈ U}forms a topology, the density topology on X, with the following property.Theorem 7.1 (Density Topology Theorem). If a density theorem holds for {K n (x) : x ∈X} and U is d-open, then every point of U is a density point of U and so U is measurable.Furthermore, a measurable set such that each point is a density point is d-open.We note that the idea of a density topology was introduced slightly earlier by Goffman([GoWa],[GNN]); see also Tall [T]. It can be traced to the work of Denjoy [Den] in1915. Recall that a function is approximately continuous in the sense of Denjoy iff it iscontinuous under the density topology: [LMZ, p. 1].Theorem 7.2 (Category-Measure Theorem – [Mar, Th. 4.11]). Suppose X is a probabilityspace and a density theorem holds for {K n (x) : x ∈ X}. A necessary and sufficientcondition that a set be nowhere dense in the d-topology is that it have measure zero.Hence a necessary and sufficient condition that a set be meagre is that it have measurezero. In particular the topological space (X, U) is a Baire space.